For 
, let 
denote the set of all subsets of 
with no two consecutive numbers. For example 
, but 
. Then 
is equal to________.
Text Solution
Verified by Experts13
(13)
To find 
, which is the number of subsets of 
with no consecutive numbers, we start by enumerating these subsets.
Let's denote the set 
as 
. The subsets of 
that meet the criteria are:
The empty set:{ 
Single-element sets: 
Two-element sets with no consecutive numbers: 
, 
Three-element set with no consecutive numbers: 
Counting these subsets, we have:
1 subset with zero elements
5 subsets with one element
6 subsets with two elements
1 subset with three elements
Adding these counts, there are 
subsets in total.
Thus, 
.
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